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Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic structure?

I am aware of partial classifications when $G$ has additional properties, but unaware of which cases are completely classified and which are still open.

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    $\begingroup$ You don't require the symplectic form to be $G$-invariant, do you? $\endgroup$
    – Francois Ziegler
    Commented Jan 19, 2014 at 19:07
  • $\begingroup$ In general, no, it does not have to be G-invariant. However, most constructions of a symplectic form on $M$ will automatically produce such an object. $\endgroup$
    – pod146
    Commented Jan 19, 2014 at 19:28
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    $\begingroup$ If a Lie group $G$ acts transitively on a symplectic manifold $M$, then $M$ is a cover of a coadjoint orbit of a central extension of $M$. If $G$ is compact connected, then its central extensions are trivial enough and its coadjoint orbits are simply connected, so $M$ is just a coadjoint orbit of $G$, hence Kähler. Which is to say, there are no new examples if the symplectic form is $G$-invariant. $\endgroup$
    – Allen Knutson
    Commented Jan 20, 2014 at 9:13
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    $\begingroup$ If you don't require $G$-invariance, maybe you can also figure out whether $G_2/SU(3)$ admits a complex structure... $\endgroup$
    – Allen Knutson
    Commented Jan 20, 2014 at 9:15

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